A thermodynamic assessment of Ni–Sb system

The Ni–Sb system was critically assessed by means of the CALculation of PHAse Diagram (CALPHAD) technique. The solution phases, Liq and (αNi), were modelled as the substitutional solutions with the Redlich–Kister equation. The intermediate phases, (γNiSb) and (βNi₃Sb), with homogeneity ranges were described respectively using three-sublattices (Sb)_1/3(Ni%,V a)_1/3(V a%,Ni)_1/3 and (Sb)_1/4(Ni%,V a)_1/2(Ni%,V a)_1/4 based on their structure features. Corresponding to the phase (βNi₃Sb), the two low-temperature phases of (δNi₃Sb) and (θNi₅Sb₂) with narrow homogeneity ranges were modelled as two-sublattice, (Ni)_3/4(Sb,Ni)_1/4 and (Ni)_5/7(Sb,Ni)_2/7. The intermetallic compound ζNiSb₂ with no homogeneity ranges was treated as stoichiometric compound. The phase ε$\epsilon$Sb was considered as pure Sb for the solubility of Ni in ε$\epsilon$Sb is very low. A set of self-consistent thermodynamic parameters of the Ni–Sb system was obtained. The optimized phase diagram and thermodynamic properties were presented and compared with experimental data.     

The Kolmogorov Expression Complexity of Logics

The purpose of the paper is to propose a completely new notion of complexity of logics in finite-model theory. It is the Kolmogorov variant of the Vardi'sexpression complexity. We define it by considering the value of the Kolmogorov complexityC(L[]) of the infinite stringL[] of all truth values of sentences ofLin . The higher is this value, the more expressive is the logicLin . If is a class of finite models, then the value ofC(L[]) over all ∈ is a measure of expressive power ofLin . Unboundedness ofC(L[])−C(L′[]) for ∈ implies nonexistence of a recursive interpretation ofLinL′. A version of this statement with complexities modulo oracles implies the nonexistence of any interpretation ofLinL′. Thus the valuesC(L[]) modulo oracles constitute an invariant of the expressive power of logics over finite models, depending on their real (absolute) expressive power, and not on the syntax. We investigate our notion for fragments of the infinitary logic ^ω_∞ω: least fixed point logic (LFP) and partial fixed point logic (PFP). We prove a precise characterization of 0–1 laws for these logics in terms of a certain boundedness condition placed onC(L[]). We get an extension of the notion of a 0–1 law by imposing an upper bound on the value ofC(L[]) growing not too fast with cardinality of , which still implies inexpressibility results similar to those implied by 0–1 laws. We also discuss classes in whichC(PFP^k[]) is very high. It appears that then PFP or its simple extension can define all the PSPACE subsets of .     

Elements of small norm in Shanks’ cubic extensions of imaginary quadratic fields

Let be an imaginary quadratic number field with ring of integers Z_k and let k(α) be the cubic extension of k generated by the polynomial f_t(x)=x³−(t−1)x²−(t+2)x−1 with tZ_k. In the present paper we characterize all elements γZ_k[α] with norms satisfying |N_k(α)/k|≤|2t+1| for |t|≥14. This generalizes a corresponding result by Lemmermeyer and Pethő for Shanks’ cubic fields over the rationals.     

Stochastic linear quadratic adaptive control for continuous-time first-order systems

In this paper, we discuss the linear quadratic (LQ) adaptive control problem for the following continuous-time first-order scalar stochastic system: dx_t = ax_t dt + bu_t dt + c dw_t, with cost function min lim sup J_t(u), where

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